Recent content by Flying_Goat

Surely the disjoint interval (0,1) with the discrete topology is 'disconnected' by defintion, but intuitively it shouldn't be. What I am confused about is that the topological properties of a space changes as you change the topology - it is not an inherent property of the space. With the usual...

Surely sets with the same cardinality are homeomorphic if we assign both of them the discrete topology. What's preventing us from doing that? For example, (0,1) and (2,3) \cup (4,5) have the same cardinality. With the natural subspace topology they are not homeomorphic - as one is connected...

Let B be the solid sphere of radius 2 in R^3. We can continuously deform everything outside of the sphere to the boundary of B via 2x/||x||. Now, if we make the open z-axis thicker then it will become a hole in the centre of the B. Similarly if we make the unit circle thicker than it will make...

Thanks for your replies. That made a lot of sense. So a homotopy equivalence allows collapsing a bunch of points into one, such as that of a disk to a point. Homeomorphism is 'stronger' in the sense that the deformation is 'bijective'.

Geometrically, what is the difference between saying 'X is homotopic equivalent to Y' and 'X is homeomorphic to Y'? I know that a homeomorphism is a homotopy equivalence, but I can't seem to visualise the difference between them. It seems to me that both of these terms are about deforming spaces...

Hi, I am stuck on two problems from Allen Hatcher's book, Algebraic Topology. Homework Statement 4. A deformation retraction in the weak sense of a space X to a subspace A is a homotopy f_t: X→X such that f_0=1_X (the identity map on X), f_1(X) \subset A, and f_t (A) \subset A for all t. Show...

Ah, I see. Thanks very much.

Hi guys, I am confused about the definition of compactification of a topological space. Suppose (X,τx) is a topological space. Define Y=X\cup{p} and a new topology τY such that U\subseteqY is open if (1) p \notin U and U\in \tauX or (2) p \in U and X-U is a compact closed subset of X...

Sorry for digging up an old thread, but I am stuck on the same problem. I let S = lim S_n so we have f(S) = f(lim S_n) = lim f(S_n) = lim S_{n+1} = S. Obvisouly S is non-empty since each f(S_n) is not empty. I am not sure if I got it right. We know that each S_n is closed and compact...

Is the map injective? Because I could have f(2n-1)=12,f(2n)=3 or f(2n-1)=1,f(2n)=23 and both of these functions would give me 0.123123...

Thanks, I had thought of your argument when I was trying to prove |P(N)|=c but it didn't work out...I can't believe that it works for this question lol. Anyway thanks for your help.

Homework Statement Let NN be the set of all functions from N to N. Prove that |NN|=c Homework Equations The Attempt at a Solution I can prove that the set of all functions from N to {0,1} has cardinality of the continuum, but i can't generalise it. Any help would be appreciated.

Thanks for your replies. I understand now.

A function defined on ℝ is continuous at x if given ε, there is a δ such that |f(x)-f(y)|<ε whenever |x-y|<δ. Does this imply that f(x+δ)-f(x)=ε? The definition only deals with open intervals so i am not sure about this. If this is not true could someone please show me a counter example for it...

Thanks for the reply Dick. So the image of the map is a subset of G, that is why if the map is not surjective then two elements must map to the same thing. Is that correct?